Linear Algebra question relating to eigenvectors

Question

Let A be an m x m positive definite symmetric matrix with eigenvalue-eigenvector pairs

$(\lambda_1,e_1),....,(\lambda_m,e_m).$ The eigenvectors are orthonormal.

Let $C = e_1e_1'+....+e_me_m'$.

Find $Ce_j$, $j=1$, ..., $m$. Also, for $x \in R^m$, find $x - Cx$.

i) $Ce_j = e_1e_1e_j+....+e_me_m'e_j$.

ii) $x - Cx = (x-e_1e_1'x+....+x-e_me_m'x)$

Not sure if I'm heading in the right direction or not.

Answer 1

It is easy to see that $$ Ce_j=e_j $$ for all $j$, and hence $C$ is the identity matrix.